First we claim that \(\Phi \)  is \(C^{\infty }\)  on \(\operatorname{St}(n,r)\) .
It is a composite of \(\usefont {T1}{cmr}{m}{n} \selectfont \fbox{Q}\text{R}\)  and \(\mathbf {Q}\mapsto \mathbf {Q}+ \gamma \nabla F(\mathbf {Q})\) . The latter is a polynomial map, only taking full-rank values by \(\textbf {C3}\)  of subsec: 1st convergence.
Meanwhile, \(\usefont {T1}{cmr}{m}{n} \selectfont \fbox{Q}\text{R}\)  is \(C^{\infty }\)  at full-rank inputs by [1]}.
So \(\Phi \)  is \(C^\infty \) .
Consequently \(\phi \)  is \(C^\infty \)  by [2]}, since \(\mathbf {Q}\mapsto \mathbf {Q}\mathbf {Q}^{\top }\)  is a quotient map from \(\operatorname{St}(n,r)\)  to \(\operatorname{Gr}(n,r)\) .
